OP-00362 Solution claimed

A universal bound on excess molecular charge

For distinct nuclei $R_1, \dots, R_M \in \mathbb{R}^3$ with charges $Z_a \in \mathbb{N}$, put $Z = \sum_a Z_a$ and let $$H_N = \sum_{j=1}^N \left( -\frac12 \Delta_{x_j} - \sum_{a=1}^M \frac{Z_a}{|x_j - R_a|} \right) + \sum_{i < j} \frac{1}{|x_i - x_j|}$$ act on $\bigwedge^N L^2(\mathbb{R}^3; \mathbb{C}^2)$. Write $E_N = \inf \sigma(H_N)$ and $E_0 = 0$; by the HVZ theorem the essential spectrum of $H_N$ is $[E_{N-1}, \infty)$. Let $N_{\max}$ be the largest $N$ for which $E_N$ is an eigenvalue of $H_N$, including an eigenvalue at the threshold $E_N = E_{N-1}$. Zhislin showed that $H_N$ has infinitely many bound states below its essential spectrum when $N \le Z$. Lieb proved $N_{\max} < 2Z + M$, without using the Pauli principle. For atoms ($M = 1$), Lieb, Sigal, Simon and Thirring proved $N_{\max} = Z + o(Z)$ as $Z \to \infty$, and Nam proved $N_{\max} < 1.22\,Z + 3Z^{1/3}$. Solovej proved the analogous bound for atoms in Hartree–Fock theory.

Problem. Is there a universal finite constant $C$ such that $$N_{\max} \le Z + CM$$ for all choices of the charges and of the nuclear positions?

Source: Lieb (1980s), and Simon (1984), Problem 10C, as reported by Lewin (2025), Open Problem 1, which states it in this form.

Imported from github.com/MColbrook/AIM.

Classification

Primary: SP — Spectral Theory
Secondary: AP, MP
Keywords: AIM Open Applied Problems, Coulomb systems, negative ions

References

@article{lewin2025,
  author  = {Lewin, Mathieu},
  title   = {Some open mathematical problems concerning charged quantum particles},
  journal = {C. R. Phys.},
  volume  = {26},
  year    = {2025},
  pages   = {369--380},
  doi     = {10.5802/crphys.249},
}
@incollection{simon1984,
  author    = {Simon, Barry},
  title     = {Fifteen problems in mathematical physics},
  booktitle = {Perspectives in Mathematics: Anniversary of Oberwolfach 1984},
  editor    = {J{\"a}ger, W. and Moser, J. and Remmert, R.},
  publisher = {Birkh{\"a}user},
  address   = {Basel},
  year      = {1984},
  pages     = {423--454},
}
@article{zhislin1960,
  author  = {Zhislin, G. M.},
  title   = {A study of the spectrum of the {S}chr{\"o}dinger operator for a system of several particles},
  journal = {Trudy Moskov. Mat. Ob{\v{s}}{\v{c}}.},
  volume  = {9},
  year    = {1960},
  pages   = {81--120},
  note    = {In Russian},
}
@article{lieb1984,
  author  = {Lieb, Elliott H.},
  title   = {Bound on the maximum negative ionization of atoms and molecules},
  journal = {Phys. Rev. A},
  volume  = {29},
  number  = {6},
  year    = {1984},
  pages   = {3018--3028},
  doi     = {10.1103/PhysRevA.29.3018},
}
@article{lsst1988,
  author  = {Lieb, Elliott H. and Sigal, Israel M. and Simon, Barry and Thirring, Walter},
  title   = {Approximate neutrality of large-{Z} ions},
  journal = {Comm. Math. Phys.},
  volume  = {116},
  number  = {4},
  year    = {1988},
  pages   = {635--644},
  doi     = {10.1007/BF01224904},
}
@article{nam2012,
  author  = {Nam, Phan Th{\`a}nh},
  title   = {New bounds on the maximum ionization of atoms},
  journal = {Comm. Math. Phys.},
  volume  = {312},
  number  = {2},
  year    = {2012},
  pages   = {427--445},
  doi     = {10.1007/s00220-012-1479-y},
}
@article{solovej2003,
  author  = {Solovej, Jan Philip},
  title   = {The ionization conjecture in {H}artree--{F}ock theory},
  journal = {Ann. of Math. (2)},
  volume  = {158},
  number  = {2},
  year    = {2003},
  pages   = {509--576},
  doi     = {10.4007/annals.2003.158.509},
}
@misc{aim-maximal-molecular-ionization,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{A universal bound on excess molecular charge}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/098-maximal-molecular-ionization.md},
  note         = {Entry 098 at commit a153f46, 2026-10-13},
}

Resolution claims

Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms v1 GitHub ↗

OpenAI

We prove that a molecule with $M$ fixed nuclei of real charges at least one and total nuclear charge $Z$ strictly binds at most $Z+CM$ electrons, where $C$ is universal. The result holds for the full nonrelativistic Coulomb Hamiltonian with two spin states and arbitrary distinct nuclear positions. For neutral atoms with integer nuclear charge $Z\ge1$, the first ionization energy …

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-19	Open → Solution claimed	The registry's curators	Uniform excess charge for Coulomb molecules and the outer radius of neutral atoms v1 recorded from math @ 1581d4c, build/main.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00362, "A universal bound on excess molecular charge", clopen.solutions.